Question

How to you solve this indefinite integral from 0 to 1? ∫(xsin(πx^2))dx

Answer 1

Answer:

$\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{\pi}$

General Formulas and Concepts:

Pre-Calculus

• Unit Circle

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Rule [Fundamental Theorem of Calculus 1]:                                     $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]:                                                         $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx$

Step 2: Integrate Pt. 1

Set variables for u-substitution.

1. Set u:                                                                                                             $\displaystyle u = \pi x^2$
2. [u] Differentiate [Basic Power Rule, Multiplied Constant]:                         $\displaystyle du = 2\pi x \ dx$

Step 3: Integrate Pt. 2

1. [Integral] Rewrite:                                                                                           $\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{2 \pi} \int\limits^1_0 {2\pi xsin(\pi x^2)} \, dx$
2. [Integral] U-Substitution:                                                                               $\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{2 \pi} \int\limits^{\pi}_0 {sin(u)} \, du$
3. [Integral] Trigonometric Integration:                                                             $\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{2 \pi}[-cos(u)] \bigg| \limits^{\pi}_0$
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus]:             $\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{2 \pi}(2)$
5. Simplify:                                                                                                         $\displaystyle \int\limits^1_0 {xsin(\pi x^2)} \, dx = \frac{1}{\pi}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit:  Integration